Approximation of Boundary Conditions of the Second and Third Types in Convection–Diffusion Equations with Applications to Environmental Hydrophysics

Introduction. A finite-difference scheme approximating a boundary value problem for a parabolic-type equation in a three-dimensional setting with boundary conditions of the first-third types is considered. This paper is a continuation of the authors’ previous works devoted to the numerical solution of one of the pressing problems of hydrophysics in shallow marine zones – the problem of transport, deposition, and transformation of suspended matter. The approximation of this lass of problems inside the domain leads to schemes converging at a rate of O(τ + h²), where h² = h²_x + h²_y + h²_z, h_x, h_y, h_z and τ are the steps of the difference grid along the spatial coordinates x, y, z and time, respectively. However, the case of boundary conditions requires careful treatment, since an inaccurate approximation may reduce the overall order of accuracy of the finite-difference scheme. The methods proposed by the authors for approximating boundary conditions ensure the convergence of the finite-difference scheme at the rate of O(τ + h²).

Materials and Methods. The authors focused on approximating third-type boundary conditions (with second-type conditions considered as a particular case). The approach is based on the central difference approximation of boundary conditions on an extended grid and the elimination of suspended matter concentration values in ghost nodes (cells).

Results. Approximations of the second- and third-type boundary conditions were constructed for a boundary value problem describing suspended matter transport. These approximations guarantee convergence of the finite-difference scheme at the rate of O(τ + h²).

Discussion. The study may be useful in convection–diffusion problems where achieving numerical solutions with acceptable accuracy is required.

Conclusion. Future research may focus on the analysis of the constructed finite-difference schemes under physically motivated constraints on the time step τ and the grid Peclet number.

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